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Augmented family: Difference between revisions

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{{Interwiki
| en = Augmented family
| de = Übermässige Temperaturen
}}
The 5-limit parent comma for the '''augmented family''' is [[128/125]], the diesis. Its [[monzo]] is {{monzo| 7 0 -3 }}, and flipping that yields {{multival| 3 0 -7 }} for the [[wedgie]]. Hence the period is 1/3 octave, and this is what is used for 5/4, the classical major third. The [[generator]] can be taken as a fifth or a semitone, and [[12edo]], with its excellent fifth, is an obvious tuning for [[5-limit]] augmented, though a sharper fifth might be preferred to go with the sharp third.
The 5-limit parent comma for the '''augmented family''' is [[128/125]], the diesis. Its [[monzo]] is {{monzo| 7 0 -3 }}, and flipping that yields {{multival| 3 0 -7 }} for the [[wedgie]]. Hence the period is 1/3 octave, and this is what is used for 5/4, the classical major third. The [[generator]] can be taken as a fifth or a semitone, and [[12edo]], with its excellent fifth, is an obvious tuning for [[5-limit]] augmented, though a sharper fifth might be preferred to go with the sharp third.


== Augmented ==
== Augmented ==
Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: 128/125
[[Comma list]]: 128/125
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Mapping generators: ~5/4, ~3
Mapping generators: ~5/4, ~3


[[POTE generator]]: ~3/2 = 706.638
[[Optimal tuning]] ([[CTE]]): ~5/4 = 1\3, ~3/2 = 701.9550


{{Val list|legend=1| 3, 12, 27, 39, 51c, 90cc }}
{{Val list|legend=1| 3, 12, 27, 39, 51c, 90cc }}
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[[Badness]]: 0.022315
[[Badness]]: 0.022315


=== Seven-limit extensions ===
=== Overview to extensions ===
The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which [[7-limit]] family member we are looking at. August adds 36/35, augene 64/63, hexe 256/245, hemiaug 245/243, and triforce 49/48. Hexe splits the period to 1/6 octave, and hemiaug the generator, giving quartertones instead of semitones.
The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which [[7-limit]] family member we are looking at. August adds [[36/35]], augene [[64/63]], hexe [[256/245]], hemiaug [[245/243]], and triforce [[49/48]]. Hexe splits the [[period]] to 1/6 octave, and hemiaug the [[generator]], giving quartertones instead of semitones.


== August ==
== August ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 36/35, 128/125
[[Comma list]]: 36/35, 128/125
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{{Multival|legend=1| 3 0 6 -7 1 14 }}
{{Multival|legend=1| 3 0 6 -7 1 14 }}


[[POTE generator]]: ~3/2 = 696.011
[[Optimal tuning]] ([[CTE]]): ~5/4 = 1\3, ~3/2 = 692.1237


{{Val list|legend=1| 9, 12, 45cd, 57cd, 69cdd }}
{{Val list|legend=1| 9, 12, 45cd, 57cd, 69cdd }}
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Mapping: [{{val| 3 0 7 -1 1 }}, {{val| 0 1 0 2 2 }}]
Mapping: [{{val| 3 0 7 -1 1 }}, {{val| 0 1 0 2 2 }}]


POTE generator: ~3/2 = 692.514
Optimal tuning (CTE): ~5/4 = 1\3, ~3/2 = 687.6853


Optimal GPV sequence: {{Val list| 9, 12, 21, 33e, 45cde }}
Optimal GPV sequence: {{Val list| 9, 12, 21, 33e, 45cde }}
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Mapping: [{{val| 3 0 7 -1 1 -3 }}, {{val| 0 1 0 2 2 3 }}]
Mapping: [{{val| 3 0 7 -1 1 -3 }}, {{val| 0 1 0 2 2 3 }}]


POTE generator: ~3/2 = 688.783
Optimal tuning (CTE): ~5/4 = 1\3, ~3/2 = 685.0836


Optimal GPV sequence: {{Val list| 9, 12f, 21, 33ef, 54bceeff }}
Optimal GPV sequence: {{Val list| 9, 12f, 21, 33ef, 54bceeff }}
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Mapping: [{{val| 3 0 7 -1 1 11 }}, {{val| 0 1 0 2 2 0 }}]
Mapping: [{{val| 3 0 7 -1 1 11 }}, {{val| 0 1 0 2 2 0 }}]


POTE generator: ~3/2 = 685.356
Optimal tuning (CTE): ~5/4 = 1\3, ~3/2 = 687.6853


Optimal GPV sequence: {{Val list| 9, 12, 21f }}
Optimal GPV sequence: {{Val list| 9, 12, 21f }}
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== Augene ==
== Augene ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 64/63, 126/125
[[Comma list]]: 64/63, 126/125
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[[Mapping]]: [{{val| 3 0 7 18 }}, {{val| 0 1 0 -2 }}]
[[Mapping]]: [{{val| 3 0 7 18 }}, {{val| 0 1 0 -2 }}]


{{Multival|legend=1|3 0 -6 -7 -18 -14}}
{{Multival|legend=1| 3 0 -6 -7 -18 -14 }}


[[POTE generator]]: ~3/2 = 709.257
[[Optimal tuning]] ([[CTE]]): ~5/4 = 1\3, ~3/2 = 709.5948


{{Val list|legend=1| 12, 27, 39d, 66cd }}
{{Val list|legend=1| 12, 27, 39d, 66cd }}
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Mapping: [{{val| 3 0 7 18 20 }}, {{val| 0 1 0 -2 -2 }}]
Mapping: [{{val| 3 0 7 18 20 }}, {{val| 0 1 0 -2 -2 }}]


POTE generator: ~3/2 = 711.177
Optimal tuning (CTE): ~5/4 = 1\3, ~3/2 = 713.5701


Optimal GPV sequence: {{Val list| 12, 15, 27e }}
Optimal GPV sequence: {{Val list| 12, 15, 27e }}
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Mapping: [{{val| 3 0 7 18 20 16 }}, {{val| 0 1 0 -2 -2 -1 }}]
Mapping: [{{val| 3 0 7 18 20 16 }}, {{val| 0 1 0 -2 -2 -1 }}]


POTE generator: ~3/2 = 712.013
Optimal tuning (CTE): ~5/4 = 1\3, ~3/2 = 716.1234


Optimal GPV sequence: {{Val list| 12f, 15, 27eff }}
Optimal GPV sequence: {{Val list| 12f, 15, 27eff }}
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Mapping: [{{val| 3 0 7 18 20 -8 }}, {{val| 0 1 0 -2 -2 4 }}]
Mapping: [{{val| 3 0 7 18 20 -8 }}, {{val| 0 1 0 -2 -2 4 }}]


POTE generator: ~3/2 = 712.609
Optimal tuning (CTE): ~5/4 = 1\3, ~3/2 = 711.9020


Optimal GPV sequence: {{Val list| 12, 15, 27e, 69bceef }}
Optimal GPV sequence: {{Val list| 12, 15, 27e, 69bceef }}
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Mapping: [{{val| 3 0 7 18 20 35 }}, {{val| 0 1 0 -2 -2 -5 }}]
Mapping: [{{val| 3 0 7 18 20 35 }}, {{val| 0 1 0 -2 -2 -5 }}]


POTE generator: ~3/2 = 709.677
Optimal tuning (CTE): ~5/4 = 1\3, ~3/2 = 712.5722


Optimal GPV sequence: {{Val list| 12f, 27e }}
Optimal GPV sequence: {{Val list| 12f, 27e }}
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Mapping: [{{val| 3 0 7 18 -4 }}, {{val| 0 1 0 -2 3 }}]
Mapping: [{{val| 3 0 7 18 -4 }}, {{val| 0 1 0 -2 3 }}]


POTE generator: ~3/2 = 714.150
Optimal tuning (CTE): ~5/4 = 1\3, ~3/2 = 713.0025


Optimal GPV sequence: {{Val list| 12e, 15, 27, 42 }}
Optimal GPV sequence: {{Val list| 12e, 15, 27, 42 }}
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; Music
; Music
* [http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Sad%20Like%20Winter%20Leaves.mp3 Sad Like Winter Leaves] by [http://soundcloud.com/cityoftheasleep/sad-like-winter-trees Igliashon Jones]
* [http://soundcloud.com/cityoftheasleep/sad-like-winter-trees ''Sad Like Winter Leaves'']{{dead link}} [http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Sad%20Like%20Winter%20Leaves.mp3 play]{{dead link}} by [[Igliashon Jones]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3 Galticeran Sonatina] by [http://soundcloud.com/joelgranttaylor/galticeran_sonatina Joel Taylor]
* [http://soundcloud.com/joelgranttaylor/galticeran_sonatina ''Galticeran Sonatina'']{{dead link}} [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3 play]{{dead link}} by [[Joel Taylor]]


== Inflated ==
== Inflated ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 28/27, 128/125
[[Comma list]]: 28/27, 128/125
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{{Multival|legend=1| 3 0 9 -7 6 21 }}
{{Multival|legend=1| 3 0 9 -7 6 21 }}


[[POTE generator]]: ~3/2 = 722.719
[[Optimal tuning]] ([[CTE]]): ~5/4 = 1\3, ~3/2 = 717.5172


{{Val list|legend=1| 3d, 15, 48bc, 63bbc, 78bbc }}
{{Val list|legend=1| 3d, 12d, 15 }}


[[Badness]]: 0.054729
[[Badness]]: 0.054729
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Mapping: [{{val| 3 0 7 -6 -4 }}, {{val| 0 1 0 3 3 }}]
Mapping: [{{val| 3 0 7 -6 -4 }}, {{val| 0 1 0 3 3 }}]


POTE generator: ~3/2 = 722.663
Optimal tuning (CTE): ~5/4 = 1\3, ~3/2 = 717.5172


Optimal GPV sequence: {{Val list| 3de, 15, 48bce, 63bbcee, 78bbcee }}
Optimal GPV sequence: {{Val list| 3de, 12de, 15 }}


Badness: 0.031171
Badness: 0.031171


== Deflated ==
== Deflated ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 21/20, 128/125
[[Comma list]]: 21/20, 128/125
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{{Multival|legend=1| 3 0 -3 -7 -13 -7 }}
{{Multival|legend=1| 3 0 -3 -7 -13 -7 }}


[[POTE generator]]: ~3/2 = 681.629
[[Optimal tuning]] ([[CTE]]): ~5/4 = 1\3, ~3/2 = 684.8473


{{Val list|legend=1| 3, 9, 21d, 30bdd }}
{{Val list|legend=1| 3, 9 }}


[[Badness]]: 0.059079
[[Badness]]: 0.059079
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Mapping: [{{val| 3 0 7 13 15 }}, {{val| 0 1 0 -1 -1 }}]
Mapping: [{{val| 3 0 7 13 15 }}, {{val| 0 1 0 -1 -1 }}]


POTE generator: ~3/2 = 680.042
Optimal tuning (CTE): ~5/4 = 1\3, ~3/2 = 679.881


Optimal GPV sequence: {{Val list| 3, 9, 21de, 30bdde }}
Optimal GPV sequence: {{Val list| 3, 9 }}


Badness: 0.037183
Badness: 0.037183


== Hexe ==
== Hexe ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 50/49, 128/125
[[Comma list]]: 50/49, 128/125
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{{Multival|legend=1| 6 0 0 -14 -17 0 }}
{{Multival|legend=1| 6 0 0 -14 -17 0 }}


[[POTE generator]]: ~3/2 = 710.963
[[Optimal tuning]] ([[CTE]]): ~28/25 = 1\6, ~3/2 = 701.9550


{{Val list|legend=1| 6, 12, 30d, 42dd, 54cdd }}
{{Val list|legend=1| 6, 12, 30d, 42dd, 54cdd }}
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Mapping: [{{val| 6 0 14 17 21 }}, {{val| 0 1 0 0 0 }}]
Mapping: [{{val| 6 0 14 17 21 }}, {{val| 0 1 0 0 0 }}]


POTE generator: ~3/2 = 714.304
Optimal tuning (CTE): ~28/25 = 1\6, ~3/2 = 701.9550


Optimal GPV sequence: {{Val list| 6, 12, 30de }}
Optimal GPV sequence: {{Val list| 6, 12, 30dee, 42ddeee }}


Badness: 0.038412
Badness: 0.038412
Line 250: Line 254:
Mapping: [{{val| 6 0 14 17 21 13 }}, {{val| 0 1 0 0 0 1 }}]
Mapping: [{{val| 6 0 14 17 21 13 }}, {{val| 0 1 0 0 0 1 }}]


POTE generator: ~3/2 = 710.005
Optimal tuning (CTE): ~28/25 = 1\6, ~3/2 = 692.4327


Optimal GPV sequence: {{Val list| 6f, 12f }}
Optimal GPV sequence: {{Val list| 6f, 12f }}
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== Hemiaug ==
== Hemiaug ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 128/125, 245/243
[[Comma list]]: 128/125, 245/243
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{{Multival|legend=1| 6 0 15 -14 7 35 }}
{{Multival|legend=1| 6 0 15 -14 7 35 }}


[[POTE generator]]: ~14/9 = 754.882 (or ~36/35 = 45.118)
[[Optimal tuning]] ([[CTE]]): ~5/4 = 1\3, ~14/9 = 752.8335 (~36/35 = 47.1665)


{{Val list|legend=1| 24, 27, 105bcc }}
{{Val list|legend=1| 24, 27 }}


[[Badness]]: 0.070463
[[Badness]]: 0.070463
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Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 56/55, 128/125, 245/243
Comma list: 56/55, 128/125, 243/242


Mapping: [{{val| 3 1 7 -1 1 }}, {{val| 0 2 0 5 5 }}]
Mapping: [{{val| 3 1 7 -1 1 }}, {{val| 0 2 0 5 5 }}]


POTE generator: ~14/9 = 754.212 (or ~36/35 = 45.788)
Optimal tuning (CTE): ~5/4 = 1\3, ~14/9 = 752.0512 (~36/35 = 47.9488)


Optimal GPV sequence: {{Val list| 24, 27e, 51ce, 78cee }}
Optimal GPV sequence: {{Val list| 24, 27e, 51ce }}


Badness: 0.038232
Badness: 0.038232
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Mapping: [{{val| 3 1 7 -1 1 13 }}, {{val| 0 2 0 5 5 -1 }}]
Mapping: [{{val| 3 1 7 -1 1 13 }}, {{val| 0 2 0 5 5 -1 }}]


POTE generator: ~14/9 = 753.750 (or ~36/35 = 46.250)
Optimal tuning (CTE): ~5/4 = 1\3, ~14/9 = 752.1284 (or ~36/35 = 47.8716)


Optimal GPV sequence: {{Val list| 24, 27e, 51ce, 78cee }}
Optimal GPV sequence: {{Val list| 24, 27e, 51ce }}


Badness: 0.030159
Badness: 0.030159
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[[File:triforce9.jpg|thumb|alt=triforce9.jpg|Lattice of triforce]]
[[File:triforce9.jpg|thumb|alt=triforce9.jpg|Lattice of triforce]]


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 49/48, 128/125
[[Comma list]]: 49/48, 128/125
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{{Multival|legend=1| 6 0 3 -14 -12 7 }}
{{Multival|legend=1| 6 0 3 -14 -12 7 }}


[[POTE generator]]: ~7/4 = 952.951
[[Optimal tuning]] ([[CTE]]): ~5/4 = 1\3, ~7/4 = 952.2948


{{Val list|legend=1| 6, 9, 15, 24, 39, 63cd, 102ccdd }}
{{Val list|legend=1| 6, 9, 15, 24, 39, 63cd, 102ccdd }}
Line 325: Line 329:
Mapping: [{{val| 3 0 7 6 8 }}, {{val| 0 2 0 1 1 }}]
Mapping: [{{val| 3 0 7 6 8 }}, {{val| 0 2 0 1 1 }}]


POTE generator: ~7/4 = 952.932
Optimal tuning (CTE): ~5/4 = 1\3, ~7/4 = 952.2495


Optimal GPV sequence: {{Val list| 6, 9, 15, 24, 39, 63cd, 102ccdd }}
Optimal GPV sequence: {{Val list| 6, 9, 15, 24, 39 }}


Badness: 0.026152
Badness: 0.026152


; Music
; Music
* [https://cityoftheasleep.bandcamp.com/track/the-triforce-of-courage-24edo The Triforce of Courage (24edo)] by [[Igliashon Jones]]
* [https://cityoftheasleep.bandcamp.com/track/the-triforce-of-courage-24edo ''The Triforce of Courage (24edo)''] by [[Igliashon Jones]] (2018)
* [http://chrisvaisvil.com/2-2-1-2-2-1-2-2-1-mode-of-15-edo/ 2-2-1-2-2-1-2-2-1 mode of 15 edo] [http://micro.soonlabel.com/15-ET/20130831_221of15.mp3 play] by [[Chris Vaisvil]]
* [http://chrisvaisvil.com/2-2-1-2-2-1-2-2-1-mode-of-15-edo/ ''2-2-1-2-2-1-2-2-1 mode of 15 edo''] [http://micro.soonlabel.com/15-ET/20130831_221of15.mp3 play] by [[Chris Vaisvil]]


==== 13-limit ====
==== 13-limit ====
Line 342: Line 346:
Mapping: [{{val| 3 0 7 6 8 4 }}, {{val| 0 2 0 1 1 3 }}]
Mapping: [{{val| 3 0 7 6 8 4 }}, {{val| 0 2 0 1 1 3 }}]


POTE generator: ~7/4 = 951.687
Optimal tuning (CTE): ~5/4 = 1\3, ~7/4 = 950.8049


Optimal GPV sequence: {{Val list| 6, 9, 15, 24, 63cdf, 87cddff }}
Optimal GPV sequence: {{Val list| 6, 9, 15, 24 }}


Badness: 0.020248
Badness: 0.020248
Line 360: Line 364:
Mapping generators: ~44/39, ~7/4
Mapping generators: ~44/39, ~7/4


POTE generator: ~7/4 = 953.358
Optimal tuning (CTE): ~44/39 = 1\6, ~7/4 = 952.5307


Optimal GPV sequence: {{Val list| 6, 24, 54cd, 78cd, 102ccddf }}
Optimal GPV sequence: {{Val list| 6, 18bd, 24 }}


Badness: 0.059160
Badness: 0.059160


== Hemiug ==
== Hemiug ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 128/125, 1323/1250
[[Comma list]]: 128/125, 1323/1250
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{{Multival|legend=1| 6 0 -9 -14 -31 -21 }}
{{Multival|legend=1| 6 0 -9 -14 -31 -21 }}


[[POTE generator]]: ~32/21 = 747.907 (or ~21/20 = 52.093)
[[Optimal tuning]] ([[CTE]]): ~5/4 = 1\3, ~32/21 = 747.9484 (~21/20 = 52.0516)


{{Val list|legend=1| 21, 24, 45c, 69cd }}
{{Val list|legend=1| 21, 24, 45c }}


[[Badness]]: 0.137764
[[Badness]]: 0.137764
Line 390: Line 394:
Mapping: [{{val| 3 1 7 14 16 }}, {{val| 0 2 0 -3 -3 }}]
Mapping: [{{val| 3 1 7 14 16 }}, {{val| 0 2 0 -3 -3 }}]


POTE generator: ~32/21 = 748.345 (or ~33/32 = 51.655)
Optimal tuning (CTE): ~5/4 = 1\3, ~32/21 = 748.2962 (~33/32 = 51.7038)


Optimal GPV sequence: {{Val list| 21, 24, 69cd, 93ccdd }}
Optimal GPV sequence: {{Val list| 21, 24 }}


Badness: 0.068062
Badness: 0.068062
Line 403: Line 407:
Mapping: [{{val| 3 1 7 14 16 13 }}, {{val| 0 2 0 -3 -3 -1 }}]
Mapping: [{{val| 3 1 7 14 16 13 }}, {{val| 0 2 0 -3 -3 -1 }}]


POTE generator: ~32/21 = 748.452 (or ~21/20 = 51.548)
Optimal tuning (CTE): ~5/4 = 1\3, ~32/21 = 748.5255 (~21/20 = 51.4745)


Optimal GPV sequence: {{Val list| 21, 24, 69cdf, 93ccddf }}
Optimal GPV sequence: {{Val list| 21, 24 }}


Badness: 0.042289
Badness: 0.042289


== Trug ==
== Trug ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 128/125, 360/343
[[Comma list]]: 128/125, 360/343
Line 418: Line 422:
Mapping generators: ~5/4, ~48/35
Mapping generators: ~5/4, ~48/35


[[POTE generator]]: ~48/35 = 501.980
[[Optimal tuning]] ([[CTE]]): ~5/4 = 1\3, ~48/35 = 498.6367


{{Val list|legend=1| 9bd, 12 }}
{{Val list|legend=1| 9bd, 12 }}
Line 425: Line 429:


== Oodako ==
== Oodako ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 128/125, 2401/2400
[[Comma list]]: 128/125, 2401/2400
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{{Multival|legend=1| 12 0 3 -28 -29 7 }}
{{Multival|legend=1| 12 0 3 -28 -29 7 }}


[[POTE generator]]: ~28/25 = 176.646
[[Optimal tuning]] ([[CTE]]): ~5/4 = 1\3, ~28/25 = 175.3586


{{Val list|legend=1| 6, 21, 27, 75c, 102ccd, 129bccd }}
{{Val list|legend=1| 6, 21, 27, 75c, 102ccd, 129bccd }}
Line 444: Line 448:
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 56/55, 128/125, 3773/3750
Comma list: 56/55, 128/125, 2401/2400


Mapping: [{{val| 3 3 7 8 10 }}, {{val| 0 4 0 1 1 }}]
Mapping: [{{val| 3 3 7 8 10 }}, {{val| 0 4 0 1 1 }}]


POTE generator: ~28/25 = 176.981
Optimal tuning (CTE): ~5/4 = 1\3, ~11/10 = 175.0533


Optimal GPV sequence: {{Val list| 6, 21, 27e }}
Optimal GPV sequence: {{Val list| 6, 21, 27e }}
Line 461: Line 465:
Mapping: [{{val| 3 3 7 8 10 12 }}, {{val| 0 4 0 1 1 -2 }}]
Mapping: [{{val| 3 3 7 8 10 12 }}, {{val| 0 4 0 1 1 -2 }}]


POTE generator: ~28/25 = 176.551
Optimal tuning (CTE): ~5/4 = 1\3, ~11/10 = 175.2524


Optimal GPV sequence: {{Val list| 6, 21, 27e }}
Optimal GPV sequence: {{Val list| 6, 21, 27e }}
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== Hemisemiaug ==
== Hemisemiaug ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 128/125, 12005/11664
[[Comma list]]: 128/125, 12005/11664
Line 478: Line 482:
{{Multival|legend=1| 12 0 18 -28 -5 42 }}
{{Multival|legend=1| 12 0 18 -28 -5 42 }}


[[POTE generator]]: ~45/28 = 855.485 (or ~36/35 = 55.485)
[[Optimal tuning]] ([[CTE]]): ~54/49 = 1\6, ~45/28 = 853.1901 (~36/35 = 53.1901)


{{Val list|legend=1| 18, 24, 42, 66c, 108bcc }}
{{Val list|legend=1| 18, 24, 42 }}


[[Badness]]: 0.210984
[[Badness]]: 0.210984
Line 491: Line 495:
Mapping: [{{val| 6 1 14 4 8 }}, {{val| 0 2 0 3 3 }}]
Mapping: [{{val| 6 1 14 4 8 }}, {{val| 0 2 0 3 3 }}]


POTE generator: ~18/11 = 855.220 (or ~36/35 = 55.220)
Optimal tuning (CTE): ~54/49 = 1\6, ~18/11 = 852.5968 (~36/35 = 52.5968)


Optimal GPV sequence: {{Val list| 24, 42e, 66ce, 108bccee }}
Optimal GPV sequence: {{Val list| 24, 42e, 66ce, 108bccee }}
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Niner gives 9 as the complexity of an otonal tetrad, tying it with augene as a temperament supported by 27edo. Niner[18], therefore, has nine such tetrads.
Niner gives 9 as the complexity of an otonal tetrad, tying it with augene as a temperament supported by 27edo. Niner[18], therefore, has nine such tetrads.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 128/125, 686/675
[[Comma list]]: 128/125, 686/675
Line 510: Line 514:
{{Multival|legend=1| 9 0 9 -21 -11 21 }}
{{Multival|legend=1| 9 0 9 -21 -11 21 }}


[[POTE generator]]: ~3/2 = 707.167
[[Optimal tuning]] ([[CTE]]): ~49/45 = 1\9, ~3/2 = 702.0044


{{Val list|legend=1| 9, 18, 27, 63c, 90cc }}
{{Val list|legend=1| 9, 18, 27, 63c, 90cc }}
Line 523: Line 527:
Mapping: [{{val| 9 0 21 11 17 }}, {{val| 0 1 0 1 1 }}]
Mapping: [{{val| 9 0 21 11 17 }}, {{val| 0 1 0 1 1 }}]


POTE generator: ~3/2 = 706.726
Optimal tuning (CTE): ~12/11 = 1\9, ~3/2 = 699.6216


Optimal GPV sequence: {{Val list| 9, 27e, 36, 63cee }}
Optimal GPV sequence: {{Val list| 9, 18e, 27e, 63cee }}


Badness: 0.034861
Badness: 0.034861
Line 536: Line 540:
Mapping: [{{val| 9 0 21 11 17 19 }}, {{val| 0 1 0 1 1 1 }}]
Mapping: [{{val| 9 0 21 11 17 19 }}, {{val| 0 1 0 1 1 1 }}]


POTE generator: ~3/2 = 706.889
Optimal tuning (CTE): ~14/13 = 1\9, ~3/2 = 700.4330


Optimal GPV sequence: {{Val list| 9, 27e, 36, 63cee }}
Optimal GPV sequence: {{Val list| 9, 18e, 27e }}


Badness: 0.024143
Badness: 0.024143
Line 548: Line 552:
[[Category:Augmented family| ]] <!-- main article -->
[[Category:Augmented family| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
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[[de:Übermässige Temperaturen]]
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